Grassmann manifolds

Revision as of 13:54, 26 November 2010

1 Introduction

Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $\F$ be the real, complex or quaternion field and $V$ a vector space over $K$ of dimension $n$ and let $k\leq n$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. {{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}} {{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. {{endthm}} Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ {{endthm}} Grassmann manifold is a homogeneous space of the general linear group. General linear group $GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$. {{beginthm|Proposition|{{cite|{Milnor&Stasheff1974}}}}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$ {{endthm}} The Grassmannians $G_1(V)$ are projective spaces, denoted $\P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_R F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection $p$ corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of $p_d$ is a sphere $S^{d-1}$. There is an embedding of the Grassmannian $G_k(\F^n)$ in the cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on $G_k(\F^n)$. Prove that there is a free action of the group $O(k,\F)$ on $V_k(\F^n)$ sucht that the orbit space is homeomorphic to $G_k(\F^n)$. Similarly for the noncompact Stiefel manifold. \end{zad} Prove that the map $p:V_k(\F^n)\to G_k(\F^n)$ is locally trivial (even a principal $O(k,\F)$-bundle), thus a fibration. They are examples of coadjoint orbits \cite{Kirillov2004} \cite{Milnor&Stasheff1974} {{beginthm|Theorem}} {{endthm}} </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $\F$ be the real, complex or quaternion field and $V$ a vector space over $K$ of dimension $n$ and let $k\leq n$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. {{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}} {{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. {{endthm}} Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ {{endthm}} Grassmann manifold is a homogeneous space of the general linear group. General linear group $GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$. {{beginthm|Proposition|{{cite|{Milnor&Stasheff1974}}}}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$ {{endthm}} The Grassmannians $G_1(V)$ are projective spaces, denoted $\P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_R F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection $p$ corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of $p_d$ is a sphere $S^{d-1}$. There is an embedding of the Grassmannian $G_k(\F^n)$ in the cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on $G_k(\F^n)$. Prove that there is a free action of the group $O(k,\F)$ on $V_k(\F^n)$ sucht that the orbit space is homeomorphic to $G_k(\F^n)$. Similarly for the noncompact Stiefel manifold. \end{zad} Prove that the map $p:V_k(\F^n)\to G_k(\F^n)$ is locally trivial (even a principal $O(k,\F)$-bundle), thus a fibration. They are examples of coadjoint orbits \cite{Kirillov2004} \cite{Milnor&Stasheff1974} {{beginthm|Theorem}} {{endthm}} </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]\F$ be the real, complex or quaternion field and $V$ a vector space over $K$ of dimension $n$ and let $k\leq n$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology.

Theorem 1.1 [{Milnor&Stasheff1974].} $G_k(V)$ is a Hausdorff, compact space.

Theorem 1.2 [{Milnor&Stasheff1974].} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$.

Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ </div>

Grassmann manifold is a homogeneous space of the general linear group. General linear group $GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$.

Proposition 1.3 [{Milnor&Stasheff1974].} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$

The Grassmannians $G_1(V)$ are projective spaces, denoted $\P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_R F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection $p$ corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of $p_d$ is a sphere $S^{d-1}$.

Tex syntax error

Tex syntax error

Tex syntax error

Tex syntax error

Tex syntax error

\end{zad}

Tex syntax error

Tex syntax error

They are examples of coadjoint orbits [Kirillov2004]

[Milnor&Stasheff1974]

2 Construction and examples

...

3 Invariants

...

4 Classification/Characterization

...

5 Further discussion

...

6 References

• [Kirillov2004] A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, American Mathematical Society, Providence, RI, 2004. MR2069175 (2005c:22001) Zbl 02121486
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [[Template:{Milnor&Stasheff1974|[{Milnor&Stasheff1974]]] {{{Milnor&Stasheff1974}}